Answers
Take the Laplace transform of both sides:
L[y'' - 4y' + 8y] = L[δ(t - 1)]
I'll denote the Laplace transform of y = y(t) by Y = Y(s). Solve for Y :
(s²Y - s y(0) - y'(0)) - 4 (sY - y(0)) + 8Y = exp(-s) L[δ(t)]
s²Y - 4sY + 8Y = exp(-s)
(s² - 4s + 8) Y = exp(-s)
Y = exp(-s) / (s² - 4s + 8)
and complete the square in the denominator,
Y = exp(-s) / ((s - 2)^2 + 4)
Recall that
L⁻¹[F(s - c)] = exp(ct) f(t)
In order to apply this property, we multiply Y by exp(2)/exp(2), so that
Y = exp(-2) • exp(-s) exp(2) / ((s - 2)² + 4)
Y = exp(-2) • exp(-s + 2) / ((s - 2)² + 4)
Y = exp(-2) • exp(-(s - 2)) / ((s - 2)² + 4)
Then taking the inverse transform, we have
L⁻¹[Y] = exp(-2) L⁻¹[exp(-(s - 2)) / ((s - 2)² + 4)]
L⁻¹[Y] = exp(-2) exp(2t) L⁻¹[exp(-s) / (s² + 4)]
L⁻¹[Y] = exp(2t - 2) L⁻¹[exp(-s) / (s² + 4)]
Next, we recall another property,
L⁻¹[exp(-cs) F(s)] = u(t - c) f(t - c)
where F is the Laplace transform of f, and u(t) is the unit step function
[tex]u(t) = \begin{cases}1 & \text{if }t \ge 0 \\ 0 & \text{if }t < 0\end{cases}[/tex]
To apply this property, we first identify c = 1 and F(s) = 1/(s² + 4), whose inverse transform is
L⁻¹[F(s)] = 1/2 L⁻¹[2/(s² + 2²)] = 1/2 sin(2t)
Then we find
L⁻¹[Y] = exp(2t - 2) u(t - 1) • 1/2 sin(2 (t - 1))
and so we end up with
y = 1/2 exp(2t - 2) u(t - 1) sin(2t - 2)
Related Questions
express (25 1/2) ^3 in simplest radical form
Answers
Step-by-step explanation:
i think this is the answer if it's wrong I'm sorry
By what number should be 6 2/9 be divided to obtain 4 2/3 ?
Answers
6(2/9)
4(2/3)
18/54
8/12
=18/54
=8/12
u divided all
Let the number should be duvided be X
Given number = 6 2/9
⇛ (6×9+2)/9
⇛ (54+2)/9
⇛ 56/9
On dividing 56/9 by X then the result
= (56/9)/X
= 56/9X
According to the given problem
The result = 4 2/3 = 14/3
⇛ 56/9X = 14/3
⇛ 9X×14 = 56×3
⇛ 9X = 56×3/14
⇛ 9X = 4×3
⇛9X = 12
⇛X = 12/9
=⇛X = 4/3 or 1 1/3
Answer↓
The required number is 4/3 or 1 1/3
Check↓
On dividing 6 2/9 by 1 1/3
⇛ 6 2/9 ÷ 1 1/3
⇛ (56/9)÷(4/3)
⇛ (56/9)×(3/4)
⇛ (56×3)/(9×4)
⇛ 14/3
⇛4 2/3
Verified.
The weight of an object on the moon is
1 of its weight on Earth. If a moon rock
weighs 203 lb on Earth, how much did
the moon rock weigh on the moon?
Answers
Answer:
The answer would be 124.4
Step-by-step explanation:
The simple answer is 124.4 lb on the moon it would way the explation behind this is that the moons gravity is 1.62m/s so dividing that by 1.62 it would be 12.4.4 lb but because of the moons twist on it's axes would make it 90 Ib
Hope this helps :)
How many litres of water can the cylinder hold??
Answers
Answer:
1.386 litres of water can the cylinder hold.
Step-by-step explanation:
For a given Cylinder
Given:radius (r) = 7 cm
height (h) = 9 cm
π = 22/7
Now,
Volume of the Cylinder
V = πr²h
V = 22/7 × 7 × 7 × 9 cm
V = 22 × 7 × 9 cm
V = 2 × 11 × 63 cm
V = 2 × 693 cm
V = 1386 cm³
Now, Divide 1386 by 1000
1386/1000 = 1.386 litre water
Thus, 1.386 litres of water can the cylinder hold.
-TheUnknownScientist 72
What are the answers pls I don’t get it
Answers
Answer:
answers are (1+0.13)x and 1.13x
Step-by-step explanation:
i did this one last year and got it right :)
1) Company A charges $55 per hour and a service charge of $250 and Company B charges $60
per hour and a service charge of $225. At what time are they equal?
Answers
Answer:
5 hours
Step-by-step explanation:
BEST ANSWER GETS BRAINLIEST!!!!
A store is having a sale on trail mix and jelly beans. For 3 pounds of trail mix and 8 pounds of jelly beans, the total cost is $31. For 5 pounds of trail mix and pounds of 2 jelly beans, the total cost is 12. Find the cost for each pound of trail mix and each pound of jelly beans.
Answers
Answer:
3x + 8y = 31
5x + 2y = 12
Let's solve your system by elimination.
Multiply the second equation by -4, then add the equations together.
(3x + 8y = 31)
−4 (5x + 2y = 12)
Becomes:
3x + 8y = 31
−20x − 8y = −48
Add these equations to eliminate y:
−17x = −17
Then solve −17x = −17 for x:
−17x = −17
−17x −17 = −17 − 17 (Divide both sides by -17)
x = 1
Now that we've found x let's plug it back in to solve for y.
Write down an original equation:
3x + 8y = 31
Substitute 1 for x in 3x + 8y = 31:
(3)(1) + 8y = 31
8y +3 = 31 (Simplify both sides of the equation)
8y + 3 − 3 = 31 − 3 (Add -3 to both sides)
8y = 28
8y8 = 288 (Divide both sides by 8)
y = 7/2
what is 2x3y if x = -2y =4
Answers
Answer:
-48
Step-by-step explanation:
2x3y
\/
2(-2) * 3(4)
\/
-4 * 12
\/
-48
Have a nice day!
I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly. (ノ^∇^)
- Heather
Answer: -48
Step-by-step explanation:
2(-2)3(4) = -4*12 = -48
how to solve subtraction of percentage from whole numbers
Answers
Please help with this other question
Answers
Answer:
pink and green same shape, but pink 7 units lower than green
Step-by-step explanation:
Both f(x) and f'(x) equations are the same except that f'(x) has the term "-7" at the end, denoting that the whole graph has been shifted down by 7 units. In other words, there is a vertical translation by -7. And since f'(x) is the pink graph, choose the option in which green and pink have the exact same shape but pink is located 7 units lower than the green.
HELP LAST ATTEMPT!!!!
Answers
Answer:
(6,-2)
Step-by-step explanation:
write a linear function for the data in each table
Answers
Answer:
19. y=5x+-1
20. y=-2.5x+4
Step-by-step explanation:
19: 9-4=5
-1+5=4 so m=5
b=-1
so y=5x+-1
20: -1-1.5=-2.5
4+-2.5=1.5
so m=-2.5
b=4
so y= -2.5x+4
Please mark me brainliest
2.(04.05)
Kelsey painted 1/4
of her bedroom in 3/7
of an hour. At this rate, how long would it take her to paint the entire room?
A.3/28 of a hour
B.28/3 of an hour
C.12/7 of an hour
D.7/17 of an hour
Answers
Answer:
1/4 in 3/7th of an hour
1/4 = 3/7
multiply by 4
1 (whole room) = 12/7 (of an hour)
Answer:
12/7 of an hour
Step-by-step explanation:
I couldnt do the asnwer earlier but here it is the person above gave the answer worked out
Mrs. Penn charges $10.00 an hour on top of a $5.00 flat fee for tutoring. Mrs. Smith charges $14.00 an hour with no flat fee. Write and solve an equation to find how many hours T takes for Mrs. Penn and Mrs. Smith to get paid the same amount.
Answers
Answer:
1.25 Hours (1 hour & 15 minutes)
Step-by-step explanation:
14x = 5 + 10x
Subtract 10x from both sides
4x = 5
Divide by 4 on both sides
x = 1.25
It will take 1 hour and 15 minutes (1.25 hours) for Mrs. Penn and Mrs. Smith to be paid the same amount.
A weather forecaster has a 60% chance of predicting the weather correctly today. If she is correct, then the probability of being correct the next day is 80%. If she is wrong, the chance of predicting incorrectly the next day is increased by 25% that she was wrong the previous day. What is the probability that the forecaster will predict the weather correctly in exactly two of the next three days?
Answers
Answer:
According to the National Weather Service, if you see a 40 percent chance of rain, "there is a 40 percent chance that rain will occur at any given point in the area."
Step-by-step explanation:
The probability that the forecaster will predict the weather correctly in exactly two of the next three days is 0.48 or 48%.
What is the chain rule in probability?For two events A and B, by chain rule, we have:
[tex]P(A \cap B) = P(B)P(A|B) = P(A)P(B|A)[/tex]
where P(A|B) is the probability of occurrence of A given that B already occurred.
A weather forecaster has a 60% chance of predicting the weather correctly today.
If she is correct, then the probability of being correct the next day is 80%.
If she is wrong, the chance of predicting incorrectly the next day is increased by 25% that she was wrong the previous day.
Let the chance of predicting the weather correctly today be A.
Let the chance of predicting the weather correctly the next day be B.
Then the probability that the forecaster will predict the weather correctly in exactly two of the next three days will be
[tex]P(A \cap B) = P(A)P(B|A) = 0.6 \times 0.8 = 0.48[/tex]
More about the chain rule of probability link is given below.
brainly.com/question/1210781
The expression on the left side of an equation is shown below. -6x+6=0 If the equation has an infinite number of solutions, which expression can be written in the box on the other side of the equation? 06 0 1 O 6x 6x6l
Answers
Answer:
I think 0
Step-by-step explanation:
is 6y +2 - y + 3 and 3 - y + 2 + 6y the same
Answers
Answer:
Step-by-step explanation:
Yes.
All you have done is changed the order. They are exactly the same.
You can sort of verify this by using
-4 + 6 - 8 + 2 = - 4
2 + 6 - 4 - 8 =
8 - 12 = - 4
Solve -x/3>5
I need help with this
Answers
x=1 becuase when 1x is it
Mr Harrison is 9 months old. How old is Mr Harrison daughter in days
Answers
Answer:
273.75 days
Step-by-step explanation:
A recipe asks for 4 1/2 pounds of chicken.
How many pounds of chicken are needed to make 1/2 of a recipe?
Express the answer in simplest form.
Enter the answer in the box.
Answers
Answer:
2/1/2 you need 2and 1/2 of chicken
A groundhog starts out in its burrow 6.2 feet below the surface of the ground. It digs straight up at a rate of 0.8 feet per
How long does it take for the groundhog to reach a depth of
-2.6 feet?
Answers
Answer:
4.5
Step-by-step explanation:
I took the test and got it right
Tt takes 20.15 minutes for the groundhog to reach a depth of -2.6 feet.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given that, a groundhog starts out in its burrow 6.2 feet below the surface of the ground.
It digs straight up at a rate of 0.8 feet per minute.
Here, rate =6.2/0.8
= 7.75 feet per minute
Now, the time is 2.6×7.75
= 20.15 minutes
Therefore, it takes 20.15 minutes for the groundhog to reach a depth of
-2.6 feet.
To learn more about the unitary method visit:
brainly.com/question/22056199.
#SPJ2
"Your question is incomplete, probably the complete question/missing part is:"
A groundhog starts out in its burrow 6.2 feet below the surface of the ground. It digs straight up at a rate of 0.8 feet per minute. How long does it take for the groundhog to reach a depth of −2.6 feet?
Use distributive property to remove the parentheses. Simplify your answer as much as possible please.
Answers
Answer:
28y^3 -35y^5
Step-by-step explanation:
= 7y^3(4) -7y^3(5y^2)
= 28y^3 -35y^5
_____
The applicable rule of exponents is ...
(y^a)(y^b) = y^(a+b)
ends of diameter: (9, 15) and (15, 7)
Answers
(
x
−
12
)
2
+
(
y
−
11
)
2
=
25
1.Amulya is x years now. Write expressions for her age 5 years ago?
Take Sarita’s Age to be y years
a) What will be her age 5 years from now?
b) What was her age 3 years now?
c) Sarita’s grandfather is 6 times her age. What is the age of her grandfather?
d)Grandmother is 2 years younger than grandfather. What is his grandmother ’s age?
e) Sarita’s father’s is age is 5 years more than 3 times Sarita’s age. What is her father’s age?
Answers
Answer:
Step-by-step explanation:
1) a = x - 5
a) z = y + 5
b) z = y - 3
c) z = 6y
d) z = 6y - 2
e) z = 3y + 5
b. A square has an area of 9 cm2 what is its side length?
Answers
Answer:
81 cm
Step-by-step explanation:
A square is equal an all sides so 9cm to the second power is 81 cm
A bag contains 8 red balls and 4 white balls. Two balls are drawn without replacement. (Enter your probabilities as fractions.) (a) What is the probability that the second ball is white, given that the first ball is red
Answers
Answer:
total marbles;12
after drawing red total counts are 7 red and 4 white 11 in total
probabilities are 4/11
A father and his young toddler are walking along the sidewalk. For every 3 steps
the father takes, the son takes 5 steps just to keep up. What is the ratio of
the number of steps the father takes to the number of steps the son takes? Add
labels to the columns of the table and place the ratio into the first row of
data. Add equivalent ratios to build a ratio table.
Answers
Answer:
Ratio 5:3
Step-by-step explanation:
5:3 equivalent fraction are 10:6, 15:9
pls help i’ll mark brainliest!!
Aiden runs 1500 meters in 8 minutes. Find the slope of the linear function that represents this relationship.
Aiden runs 187.5 meters perminute.
Aiden runs 180 meters per minute.
Aiden runs 200 meters per minute.
Aiden runs 12 meters per minute.
Answers
Explanation: Literally divide the two numbers
The slope of the linear function that represents this relationship is Aiden runs 187.5 meters per minute.
What is Slope?A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,
m = Δy/Δx where, m is the slope
Given:
Aiden runs 1500 meters in 8 minutes.
We know slope= Rise/ Run
So, slope= 1500 /8
slope= 187.5
Hence, the slope of the linear function that represents this relationship is Aiden runs 187.5 meters per minute.
Learn more about slope here:
https://brainly.com/question/3605446
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Help on this fast?? Thank you
Answers
Answer:
The answer is 17.5 dollars an hour, if you were to divide 70 by 4 that is what you would get as a result.
Step-by-step explanation:
From a large number of actuarial exam scores, a random sample of scores is selected, and it is found that of these are passing scores. Based on this sample, find a confidence interval for the proportion of all scores that are passing. Then find the lower limit and upper limit of the confidence interval. Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places.
Answers
Supposing 60 out of 100 scores are passing scores, the 95% confidence interval for the proportion of all scores that are passing is (0.5, 0.7).
The lower limit is 0.5.The upper limit is 0.7.In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the z-score that has a p-value of [tex]\frac{1+\alpha}{2}[/tex].
60 out of 100 scores are passing scores, hence [tex]n = 100, \pi = \frac{60}{100} = 0.6[/tex]
95% confidence level
So [tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so [tex]z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6 - 1.96\sqrt{\frac{0.6(0.4)}{100}} = 0.5[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6 + 1.96\sqrt{\frac{0.6(0.4)}{100}} = 0.7[/tex]
The 95% confidence interval for the proportion of all scores that are passing is (0.5, 0.7).
The lower limit is 0.5.The upper limit is 0.7.A similar problem is given at https://brainly.com/question/16807970